Statistical Mechanics - Physics at Oregon State University

IV CONTENTS
4.3 Gas of poly-atomic molecules. . . . . . . . . . . . . . . . . . . . . 77
4.4 Degenerate gas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.5 Fermi gas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
4.6 Boson gas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
4.7 Problems for chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . 87
5 Fermions and Bosons 89
5.1 Fermions in a box. . . . . . . . . . . . . . . . . . . . . . . . . . . 89
5.2 Bosons in a box. . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
5.3 Bose-Einstein condensation. . . . . . . . . . . . . . . . . . . . . . 113
5.4 Problems for chapter 5 . . . . . . . . . . . . . . . . . . . . . . . . 115
6 Density matrix formalism. 119
6.1 Density operators. . . . . . . . . . . . . . . . . . . . . . . . . . . 119
6.2 General ensembles. . . . . . . . . . . . . . . . . . . . . . . . . . . 123
6.3 Maximum entropy principle. . . . . . . . . . . . . . . . . . . . . . 125
6.4 Equivalence of entropy definitions for canonical ensemble. . . . . 134
6.5 Problems for chapter 6 . . . . . . . . . . . . . . . . . . . . . . . . 136
7 Classical statistical mechanics. 139
7.1 Relation between quantum and classical mechanics. . . . . . . . . 139
7.2 Classical formulation of statistical mechanical properties. . . . . 142
7.3 Ergodic theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
7.4 What is chaos? . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
7.5 Ideal gas in classical statistical mechanics. . . . . . . . . . . . . . 149
7.6 Normal systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
7.7 Quadratic variables. . . . . . . . . . . . . . . . . . . . . . . . . . 151
7.8 Effects of the potential energy. . . . . . . . . . . . . . . . . . . . 152
7.9 Problems for chapter 7 . . . . . . . . . . . . . . . . . . . . . . . . 154
8 Mean Field Theory: critical temperature. 157
8.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
8.2 Basic Mean Field theory. . . . . . . . . . . . . . . . . . . . . . . . 161
8.3 Mean Field results. . . . . . . . . . . . . . . . . . . . . . . . . . . 164
8.4 Density-matrix approach (Bragg-Williams approximation. . . . . 168
8.5 Critical temperature in different dimensions. . . . . . . . . . . . . 177
8.6 Bethe approximation. . . . . . . . . . . . . . . . . . . . . . . . . 181
8.7 Problems for chapter 8 . . . . . . . . . . . . . . . . . . . . . . . . 188
9 General methods: critical exponents. 191
9.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
9.2 Integration over the coupling constant. . . . . . . . . . . . . . . . 192
9.3 Critical exponents. . . . . . . . . . . . . . . . . . . . . . . . . . . 197
9.4 Relation between susceptibility and fluctuations. . . . . . . . . . 203
9.5 Exact solution for the Ising chain. . . . . . . . . . . . . . . . . . 205
9.6 Spin-correlation function for the Ising chain. . . . . . . . . . . . . 210